The mass, which would have to surround our Universe (in your pull-it-apart idea), would be equivalent to a shell.
Within a shell, gravity (from that shell) is effectively zero as it is cancelled out by the attraction from all around.
For example, if you dig into the Earth, you can discount the gravitational effect of the Earth above your height, all around the Earth, in calculating what gravity you experience; until at the centre of the Earth you experience no gravity. (More or less, the earth isn't uniform).
So I'm not sure your idea holds.
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